On page 9 of Convex Optimization Theory, a function $f: X \rightarrow [-\infty, \infty]$, $X\subset \mathbb{R}^n$ is defined to be closed if its epigraph $\mathrm{epf}(f)$ defined as: $$ \mathrm{epi}(f):= \{(x, w) |x\in X, w\in\mathbb{R}, f(x) \leq w \} $$ is closed. However, it is not clear whether $\mathrm{epf}(f)$ is required to be closed with respect to the topology on $\mathbb{R}^n\times \mathbb{R}$ or with respect to the subspace topology of $X\times \mathbb{R}$. I am guessing the former because it is mentioned at the end of page 10 that this definition is domain independent. I just want to make sure.
Thank you!
